Significant works have been done on the upper bound of the gap between consecutive primes by various authors without assuming an unproved hypothesis. Hoheisel was the first to show in 1930 the existence of a constant \(\delta >0\)  (mainly \(\delta = 1/33000\) ) such that \(p_{n+1}-p_n = O({p_n}^{1-\delta })\)  [1]}. Heilbronn [2]}, and Tchudakoff [3]}, both improved on the value of \(\delta \) . Ingham [4]} made a significant progress that contributed to the first solutions surrounding the problem of existence of a prime between two consecutive cubes.
